Boundary output feedback stabilization of a cascade of N heat equations

TL;DR

This paper develops boundary output feedback stabilization methods for a cascade of N heat equations with boundary coupling, using spectral analysis and modal controllability, applicable to different spectral configurations.

Contribution

It introduces a stabilization approach for coupled heat equations with boundary control, handling both disconnected and multiplicity eigenvalue cases, via spectral and modal analysis.

Findings

Stabilization achieved in L^2 and H^1 norms.

Spectral analysis confirms controllability and observability.

Generalized eigenvectors form a Riesz basis, ensuring stability.

Abstract

This paper solves the problem of output feedback stabilization for a cascade of N heat equations that are coupled at the boundary, the input being a scalar boundary control applied to the first heat equation of the cascade, and the scalar output being either a distributed or a pointwise in-domain measurement done on the last equation of the cascade. Two different configurations are studied in details. The first one consists of a cascade of N heat equations with totally disconnected spectra. The second one consists of a cascade of N identical heat equations, inducing eigenvalues of multiplicity N . In both cases, the problem is solved thanks to a spectral analysis and a study of the modal controllability and observability properties. The key point is that the generalized eigenvectors form a Riesz basis of the state space. The stabilization property is established in L^2 and H^1 norms.